428 BELL SYSTEM TECHNICAL JOURNAL 



particular condition of sustained oscillation, vary slightly with the 

 magnitude of the oscillations. 3 With this assumption the frequencies 

 of the components of the electromotive force reduce to a)„ ± wco,,,. 



If the electric circuit is resonant at any of these frequencies, we 

 may as above neglect the currents at other frequencies. In the 

 absence of any resonance, if the constant of non-linearity, K, is 

 sufficiently small, the amplitudes will decrease so rapidly with in- 

 creasing n that we may neglect all for which n is greater than unity. 

 In the interests of simplicity we shall make all of these assumptions. 

 We have then in addition to i„ and F,„ the currents, 



i^ = ^ [e'(c-.(+0») _|_ g---:(a,.,+e.)-]^ (-11) 



ij = y [g'("./(+o^) + e-'(o>.it+B.i)j (12) 



Forced Oscillations; Impedance Solution 



We wish to set up the equations of motion in terms of the applied 

 forces, the velocities and currents at the various frequencies, and the 

 properties of the system, as expressed in terms of its linear impedances 

 and the constant of non-linearity, K. For each frequency, we equate 

 whatever applied force there may be to the sum of the restoring forces 

 due to the system. These consist of a component given by the 

 product of the velocity or current by the impedance for the particular 

 frequency, and other terms due to the combination of pairs of the 

 other frequencies. To find these latter components, we integrate (6), 

 (8), (11) and (12) with respect to time, substitute the resulting dis- 

 placements in the last terms of (2) and (3), and select the components 

 of the four significant frequencies, for insertion in their appropriate 

 equations. Once these components are obtained, we may, since the 

 remainder of the system is linear, safely revert to the conventional 

 use of exponentials, so that the factor e'"' may be divided out for each 

 equation. The final result is 



ZJ,e^^»o+<P<,) + ^Ijuli gue.-e.,) _ ^1j^ ^,(0,n+Ba) = Eg, (13) 



Z„F„e'('''"+^"'> + ^^^ g'(9.-9») + ^LiIj g>(0„-e,) ^ F„,e'^"', (14) 



^j^g,(9.,+^,) _ ^^llhl eue,+e,n) = q, (15) 



KT V 



Zdde^'''^^'^ + " ^.(e^-fl".) = 0. (16) 



( 



